Physical Principles of Electron Microscopy: An Introduction to TEM ...

Electron Optics 51
From Eq. (2.7), the focal length of the lens can be written as f = AE0 where
A is independent of electron energy, and taking derivatives gives: �f = A �E0
= (f/E0) �E0 . The loss of spatial resolution due to chromatic aberration is
therefore:
rc � �f (�E0/E0) (2.20)
More generally, a coefficient of chromatic aberration Cc is defined by the
equation:
rc � �Cc (�E0 /E0) (2.21)
Our analysis has shown that Cc = f in the thin-lens approximation. A more
exact (thick-lens) treatment gives Cc slightly smaller than f for a weak lens
and typically f/2 for a strong lens; see Fig. 2-13. As in the case of spherical
aberration, chromatic aberration cannot be eliminated through lens design
but is minimized by making the lens as strong as possible (large focusing
power, small f ) and by using an angle-limiting aperture (restricting �). High
electron-accelerating voltage (large E0) also reduces the chromatic effect.
Axial astigmatism
So far we have assumed complete axial symmetry of the magnetic field that
focuses the electrons. In practice, lens polepieces cannot be machined with
perfect accuracy, and the polepiece material may be slightly inhomogeneous,
resulting in local variations in relative permeability. In either case, the
departure from cylindrical symmetry will cause the magnetic field at a given
radius r from the z-axis to depend on the plane of incidence of an incoming
electron (i.e., on its azimuthal angle �, viewed along the z-axis). According
to Eq. (2.9), this difference in magnetic field will give rise to a difference in
focusing
power, and the lens is said to suffer from axial astigmatism.
Figure 2.15a shows electrons leaving an on-axis object point P at equal
angles to the z-axis but traveling in the x-z and y-z planes. They cross the
optic axis at different points, Fx and Fy , displaced along the z-axis. In the
case of Fig. 2-15a, the x-axis corresponds to the lowest focusing power and
the perpendicular y-direction corresponds to the highest focusing power.
In practice, electrons leave P with all azimuthal angles and at all angles
(up to �) relative to the z-axis. At the plane containing Fy , the electrons lie
within a caustic figure that approximates to an ellipse whose long axis lies
parallel to the x-direction. At Fx they lie within an ellipse whose long axis
points in the y-direction. At some intermediate plane F, the electrons define a
circular disk of confusion of radius R, rather than a single point. If that plane
is used as the image (magnification M), astigmatism will limit the point
resolution to a value R/M.